Editing Uncountable set (section)

==Without the axiom of choice==
{{main|Dedekind-infinite set}}
Without the [[axiom of choice]], there might exist cardinalities [[Comparability|incomparable]] to <math>\aleph_0</math> (namely, the cardinalities of [[Dedekind-finite]] infinite sets). Sets of these cardinalities satisfy the first three characterizations above, but not the fourth characterization. Since these sets are not larger than the natural numbers in the sense of cardinality, some may not want to call them uncountable.

If the axiom of choice holds, the following conditions on a cardinal <math>\kappa</math> are equivalent:
*<math>\kappa \nleq \aleph_0;</math>
*<math>\kappa > \aleph_0;</math> and
*<math>\kappa \geq \aleph_1</math>, where <math>\aleph_1 = |\omega_1 |</math> and <math>\omega_1</math> is the least [[initial ordinal]] greater than <math>\omega.</math>

However, these may all be different if the axiom of choice fails. So it is not obvious which one is the appropriate generalization of "uncountability" when the axiom fails. It may be best to avoid using the word in this case and specify which of these one means.